Continuity: Length of Sullivan's cigarette and ash during their lengthy discussion
Audio/visual unsynchronized: When Florence is playing the piano in the bedroom, the notes she plays are too high for the position on the keyboard.
Continuity: As Temple discusses Princess Ida, he removes all of his makeup with cold cream. At the end of the scene, he removes heavy black eyeliner that wasn't there a moment ago.
Factual errors: When Gilbert enters the night's take for Princess Ida in his ledger, the date can be seen to be listed as Monday, 10 June, 1884. That date actually occurred on a Tuesday in that year.
Anachronisms: Gilbert sarcastically suggests that Sullivan, wanting to write a more serious opera, might try to collaborate with "Mr. Ibsen in Oslo." At the period in which the movie is set, the city was named Christiania.
Continuity: When Dr. Gilbert is complaining about the telephone, he sits down twice.
Factual errors: The Japanese exhibition that Gilbert and Lucy attend did not open until after Gilbert had started work on "The Mikado". Nor did Gilbert purchase a Japanese sword from said exhibition.
Errors made by characters (possibly deliberate errors by the filmmakers): On opening night of "The Mikado", Mr. Gilbert stands outside Mr. Grosssmith's dressing room door and wishes him "good luck". As any theater professional is well aware, it is, in the industry, considered extremely bad luck to wish an actor "good luck" prior to a performance - the proper expression is "break a leg", which was certainly well known and in common use during Gilbert's era. It is therefore highly unlikely and implausible that a savvy and experienced show business professional like Mr. Gilbert would defy hundreds of years of theater tradition and dare to say "good luck" to an actor on opening night. (Of course, Mr. Gilbert was extremely nervous at the time so there's always the possibility that he slipped up and said it inadvertently.)